Line Intersection Calculator
Find where two lines intersect and calculate the angle between them
Line 1: y = m₁x + b₁
Line 2: y = m₂x + b₂
Graph
Step-by-Step Solution
Line Intersection
Finding Intersection Point
To find where two lines intersect, set their equations equal and solve for x:
m₁x + b₁ = m₂x + b₂
x = (b₂ - b₁) / (m₁ - m₂)
Then substitute x back into either equation to find y.
Three Possible Cases
- One intersection point: Lines have different slopes (most common case)
- No intersection (parallel): Lines have same slope but different y-intercepts
- Infinite intersections (same line): Lines have same slope and same y-intercept
Angle Between Lines
The acute angle θ between two lines with slopes m₁ and m₂ is:
Applications
- Finding collision points in physics and gaming
- Solving systems of linear equations
- Determining break-even points in economics
- Traffic intersection design
- Computer graphics and ray tracing
Frequently Asked Questions
How do I find where two lines intersect?
Set the equations equal to each other and solve for x, then substitute back to find y.
What if the lines are parallel?
Parallel lines have the same slope but different y-intercepts. They never intersect, so there's no intersection point.
Can two lines intersect at more than one point?
No, except when they're the same line (infinite intersection points). Two distinct lines can only intersect at one point or not at all.
How do I find the angle between two lines?
Use the formula: tan(θ) = |m₁ - m₂| / |1 + m₁m₂|, where m₁ and m₂ are the slopes.
What if one line is vertical?
A vertical line (x = c) can't be written as y = mx + b. To find intersection with another line, substitute x = c into the other equation to find y.
What does it mean if lines are coincident?
Coincident means the lines are identical - they have the same slope and y-intercept. Every point on one line is also on the other.
Can perpendicular lines have any angle?
Perpendicular lines always intersect at exactly 90°. If the angle is different, they're not perpendicular.
How do I check my answer?
Substitute the intersection point into both original equations. If both equations are satisfied, your answer is correct.